Non-existence of a parametrically compatible metric to a complete geodesible vector field on $\mathbb{R}^2\setminus\{p,q\}$

Question

Inspired by this answer to the question entitled "Possible isometry groups of open manifolds" we ask the following question:

Is there a complete vector field $X$ on $\mathbb{R}^2\setminus\{p,q\}$ whose foliation is a geodesible foliation but there is no a Riemannian metric $g$ on $\mathbb{R}^2\setminus\{p,q\}$ such that all trajectories of $X$ are length parametrized geodesics? ($|X|_g=1$)